312. Burst Balloons

题目：

Given

n

balloons, indexed from

0

to

n-1

. Each balloon is painted with a number on it represented by array

nums

. You are asked to burst all the balloons. If the you burst balloon

i

you will get

nums[left] * nums[i] * nums[right]

coins. Here

left

and

right

are adjacent indices of

i

. After the burst, the

left

and

right

then becomes adjacent.

Find the maximum coins you can collect by bursting the balloons wisely.

Note:
(1) You may imagine

nums[-1] = nums
= 1

. They are not real therefore you can not burst them.
(2) 0 ≤

n

≤ 500, 0 ≤

nums[i]

≤ 100

Example:

Given

[3, 1, 5, 8]

Return

167

nums = [3,1,5,8] --> [3,5,8] -->   [3,8]   -->  [8]  --> []
coins =  3*1*5      +  3*5*8    +  1*3*8      + 1*8*1   = 167

链接： http://leetcode.com/problems/burst-balloons/

题解：

射气球游戏，射中气球以后得分是nums[i] * nums[i - 1] * nums[i - 2]，之后左右两边气球相连。 这道题思路也很绕，看了dietpepsi的解答也很模糊，跟买卖股票with cooldown一样。方法应该是用dp或者divide and conquer，大概想法是每burst掉一个气球，我们做一遍2d dp。代码并不长，所以我把答案背下来了...擦。理解交给二刷了。

Time Complexity - O(n3)， Space Complexity - O(n2)

public class Solution {
public int maxCoins(int[] orgNums) {
if(orgNums == null || orgNums.length == 0) {
return 0;
}
int len = orgNums.length + 2;
int[] nums = new int[len];
nums[0] = nums[len - 1] = 1;            // boundary
for(int i = 0; i < orgNums.length; i++) {
nums[i + 1] = orgNums[i];
}
int[][] dp = new int[len][len];

for(int i = 1; i < len; i++) {          // first balloon
for(int lo = 0; lo < len - i; lo++) {   // left part
int hi = lo + i;                    // right part boundary
for(int k = lo + 1; k < hi; k++) {
dp[lo][hi] = Math.max(dp[lo][hi], nums[lo] * nums[k] * nums[hi] + dp[lo][k] + dp[k][hi]);
}
}
}

return dp[0][len - 1];
}
}

Reference:
https://leetcode.com/discuss/72216/share-some-analysis-and-explanations https://leetcode.com/discuss/72186/c-dynamic-programming-o-n-3-32-ms-with-comments https://leetcode.com/discuss/72215/java-dp-solution-with-detailed-explanation-o-n-3 https://leetcode.com/discuss/72683/my-c-code-dp-o-n-3-20ms https://leetcode.com/discuss/73288/python-dp-n-3-solutions https://leetcode.com/discuss/72802/share-my-both-dp-and-divide-conquer-solutions https://leetcode.com/discuss/73924/my-36ms-c-solution